| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > f1orn | Structured version Visualization version GIF version | ||
| Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
| Ref | Expression |
|---|---|
| f1orn | ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dff1o2 6853 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹)) | |
| 2 | eqid 2737 | . . 3 ⊢ ran 𝐹 = ran 𝐹 | |
| 3 | df-3an 1089 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹)) | |
| 4 | 2, 3 | mpbiran2 710 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
| 5 | 1, 4 | bitri 275 | 1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ◡ccnv 5684 ran crn 5686 Fun wfun 6555 Fn wfn 6556 –1-1-onto→wf1o 6560 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-ex 1780 df-cleq 2729 df-ss 3968 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: f1f1orn 6859 infdifsn 9697 efopnlem2 26699 cycpmcl 33136 |
| Copyright terms: Public domain | W3C validator |